Semidiscrete Finite Element Research Articles

Interpolated coefficients stabilizer-free weak Galerkin method for semilinear parabolic convection–diffusion problem

We continue our effort in Li et al. (2024) to explore an interpolated coefficients stabilizer-free weak Galerkin finite element method (IC SFWG-FEM) to solve a one-dimensional semilinear parabolic convection–diffusion equation. Due to the introduction of interpolated coefficients and the design without stabilizers, this method not only possesses the capability of approximating functions and sparsity in the stiffness matrix, but also reduces the complexity of analysis and programming. Theoretical analysis of stability for the semi-discrete IC SFWG finite element scheme is provided. Moreover, numerical experiments are carried out to demonstrate the effectivity and stability.

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

Finite element analysis of extended Fisher-Kolmogorov equation with Neumann boundary conditions

This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains R⊂Rd, where d≤3. Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings.

Error estimates of semidiscrete finite element approximation for minimal time impulse control problems

Error estimates of semidiscrete finite element approximation for minimal time impulse control problems

A novel finite element approximation of anisotropic curve shortening flow

We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here, the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to, for instance, geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal H^1 -error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.

Two‐grid mixed finite element method for two‐dimensional time‐dependent Schrödinger equation

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in ‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in ‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method.

A priori error estimates of finite volume element method for bilinear parabolic optimal control problem

<abstract><p>In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $ O(h^{3/2}+\triangle t) $ in the sense of $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. Finally, a numerical experiment is presented to test these theoretical results.</p></abstract>

A positivity-preserving numerical algorithm for stochastic age-dependent population system with Lévy noise in a polluted environment

This paper develops a model for describing a stochastic age-dependent population system (SADPS) with Lévy noise in a polluted environment. As the model includes a nonlinear drift term and Lévy noise, it is difficult, if not impossible, to derive an analytical solution of the model. This study thus aims at developing a numerical algorithm that is based on a modified truncated Euler-Maruyama (EM) method and positivity preserving. We first study global positivity for the solution of the SADPS model in a polluted environment. Subsequently, we present a semi-discrete Galerkin finite element method in an age variable, and construct a full-discrete numerical approximation using modified truncated EM scheme on time to preserve positivity. Under certain suitable regularity conditions, estimates of convergence error of the full-discrete and a semi-discrete scheme are presented. Finally, a numerical example is given to illustrate the theoretical results.

Stability and Convergence Analysis of a Domain Decomposition FE/FD Method for Maxwell’s Equations in the Time Domain

Stability and convergence analyses for the domain decomposition finite element/finite difference (FE/FD) method are presented. The analyses are developed for a semi-discrete finite element scheme for time-dependent Maxwell’s equations. The explicit finite element schemes in different settings of the spatial domain are constructed and a domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.

Finite element analysis of nonlinear reaction–diffusion system of Fitzhugh–Nagumo type with Robin boundary conditions

In this paper, we investigate the numerical analysis of Fitzhugh–Nagumo (FHN) reaction–diffusion equations. The properties of numerical solutions of a semi-discrete and fully-practical piecewise linear finite element technique are provided. Moreover, for a semi-discrete and fully discrete finite element approximation, we establish a priori estimates and error bounds. We also introduce the results of some numerical examples in one and two dimensions, which confirm the theoretical findings of this paper.

Multipoint flux mixed finite element method for parabolic optimal control problems

<abstract><p>In this paper, we research semi-discrete multipoint flux mixed finite element (MFMFE) method for parabolic optimal control problem (OCP). The state and co-state variables are approximated by the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element (MFE) spaces and the control is approximated by piecewise constant. The advantage of this type of mixed element method is that it can decouple the state and adjoint state variables as cell-centered difference schemes rather than to solve saddle point algebraic equations. Error estimates and convergence orders are derived rigorously for state and control variables. Finally, a numerical example is given to confirm our theoretical analysis.</p></abstract>

Numerical analysis for a system coupling curve evolution attached orthogonally to a fixed boundary, to a reaction–diffusion equation on the curve

AbstractWe consider a semidiscrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain , such that the curve meets the boundary orthogonally, and the forcing is a function of the solution of a reaction–diffusion equation that holds on the evolving curve. We prove optimal order error bounds for the resulting approximation and present numerical experiments.

A convergent evolving finite element algorithm for Willmore flow of closed surfaces

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

Analysis of a Galerkin finite element method for the Maxwell–Schrödinger system under temporal gauge

Abstract This paper is concerned with the numerical solution of the Maxwell–Schrödinger system under the temporal gauge, which describes light–matter interactions. We first propose a semidiscrete finite element scheme for the system and establish stability estimates for the finite element solution. Due to the lack of control over its divergence we cannot get $\textbf{H}^{1}$$a \;priori$ estimates for the vector potential, making it difficult to obtain error estimates by usual techniques. We apply an exhaustion argument to overcome this difficulty and derive error estimates for the finite element approximation. An energy-conserving time-stepping scheme is proposed to solve the semidiscrete system.

Finite element analysis for a diffusion equation on a harmonically evolving domain

Abstract We study convergence of the evolving finite element semidiscretization of a parabolic partial differential equation on an evolving bulk domain. The boundary of the domain evolves with a given velocity, which is then extended to the bulk by solving a Poisson equation. The numerical solution to the parabolic equation depends on the numerical evolution of the bulk, which yields the time-dependent mesh for the finite element method. The stability analysis works with the matrix–vector formulation of the semidiscretization only and does not require geometric arguments, which are then required in the proof of consistency estimates. We present various numerical experiments that illustrate the proven convergence rates.

Maximal regularity of multistep fully discrete finite element methods for parabolic equations

AbstractThis article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

On motion by curvature of a network with a triple junction

We numerically study the planar evolution by curvature flow of three parametrised curves that are connected by a triple junction in which conditions are imposed on the angles at which the curves meet. One of the key problems in analysing motion of networks by curvature law is the choice of a tangential velocity that allows for motion of the triple junction, does not lead to mesh degeneration, and is amenable to an error analysis. Our approach consists in considering a perturbation of a classical smooth formulation. The problem we propose admits a natural variational formulation that can be discretized with finite elements. The perturbation can be made arbitrarily small when a regularisation parameter shrinks to zero. Convergence of the new semi-discrete finite element scheme including optimal error estimates are proved. These results are supported by some numerical tests. Finally, the influence of the small regularisation parameter on the properties of scheme and the accuracy of the results is numerically investigated.

Convergence of Dziuk's Semidiscrete Finite Element Method for Mean Curvature Flow of Closed Surfaces with High-order Finite Elements

Dziuk's surface finite element method (FEM) for mean curvature flow has had a significant impact on the development of parametric and evolving surface FEMs for surface evolution equations and curva...

A Finite Element Splitting Method for a Convection-Diffusion Problem

Abstract For a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k / m {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

On the numerical solution of two-dimensional Rosenau–Burgers (RB) equation

Our purpose is to solve numerically the nonlinear evolutionary problem called Rosenau–Burgers (RB) equation. First, we have derived error estimates of semidiscrete finite element method for the approximation of the Rosenau–Burgers problem. For a second-order accuracy in time, we propose the Galerkin–Crank–Nicolson fully discrete method. Second, a linearized difference scheme for the Rosenau–Burgers equation is considered. It is proved that the proposed difference scheme is uniquely solvable, and the method is shown to be second-order convergent both in time and space in maximum norm. Finally, some numerical experiments are given to demonstrate the validity and accuracy of our linearized difference scheme.